This is a note of real and complex analysis chapter 2.

Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear **functionals**, not combination, of a continuous function space on compact set (\(C\)) (\(\Lambda f\)) represents the integration of the function (\(\int f du\)) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let \(\Lambda\) be a positive linear functional on \(C_c(X)\). Then there exist a \(\sigma-algebra\) in \(X\) which contains all Borel sets in \(X\), and there exists a unique positive measure \(mu\) on \(\mathfrak{M}\) which represents \(\Lambda\) in the sense that (a) \(\Lambda f = \int f d \mu\) for every \(f \in C_c(X)\) and following additional properties:

(b) \(\mu(K) < \infty\) for every compact set \(K \subset X\).

(c) For every \(E \in \mathfrak{M}\), we have \[ \mu(E) = inf\{\mu(V): E in V, V open\} \].

(d) The relation \[\mu(E)=sup\{\mu(K): K \in E, K compact\}\]

holds for every open set \(E\), and for every \(E \in M\) with \(\mu(E) < \infty\).

(e) If \(E \in \mathfrak{M}, A subset E\), and \(\mu(E) = 0\), then \(A \in \mathfrak{M}\).

The Riesz theorem is about linear functional \(\Lambda\) is equivalently replaced with choosing measure \(\mu(E)=sup\{\Lambda f: f \prec V\}\). Note \(sup \{\int^1_0 f(x)dx = \Lambda f: f \prec V, V (0,1) \} = 1\). The notion of \(\prec\) include \(0 \le f \le 1\).

I confused \(C_c(X)\)